research article plastic constitutive model and analysis

Hindawi Publishing CorporationISRNMechanical EngineeringVolume 2013, Article ID 490750, 6 pageshttp://dx.doi.org/10.1155/2013/490750

Research ArticlePlastic Constitutive Model and Analysis of Flow Stress of40Cr Quenched and Tempered Steel

Wang Xiao-qiang,1 Zhu Wen-juan,1 Cui Feng-kui,1 and Li Yu-xi2

1 School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang 471003, China2 School of Mechanical and Precision Instrument Engineering, Xi An University of Technology, Xi An 710048, China

Correspondence should be addressed to Zhu Wen-juan; [email protected]

Received 6 June 2013; Accepted 10 July 2013

Academic Editors: A. Z. Sahin, A. Szekrenyes, and X. Yang

Copyright © 2013 Wang Xiao-qiang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

To solve the problem of the accuracy of the numerical simulations of cold rolling, the thermomechanical responses of 40Crunder uniaxial compression loading are presented. The strain rates include quasistatic (0.004 s−1) at temperature of 293K anddynamic loading regime (632 s−1∼5160 s−1) at temperature regime (293K∼673K). Significant strain rate and temperature sensitivityare measured. Based on these observations, the Johnson-Cook phenomenological constitutive model is proposed to predict themechanical behavior of the 40Cr over wide ranges of strain rate and temperature. The solution process of the equation parametersis given. Correlationswith this Johnson-Cookmodel are shown very close to the observed responses. Importantmaterial parametersare provided to the application of numerical analysis in project.

1. Introduction

High-speed cold rolling is an advanced cold bulk metalforming technology. During the forming process, the work-piece is interruptedly struck by high-speed rolls, and plasticdeformation of the workpiece is gradually achieved [1, 2].40Cr quenched and tempered steel is the main structurematerial used in high-speed cold rolling technology becauseof its good strength and ductility combination. During high-speed cold rolling forming, 40Cr quenched and temperedsteel is subjected to high strain rates (about 900 s−1∼4000 s−1)and large deformation. Numerical analysis is a critical toolfor understanding the complex deformation mechanics thatoccur during cold rolling forming processes. Confidence inthe numerical analysis of formability depends on the accu-racy of the constitutive model describing the behavior ofthe material. Therefore, there is a good need to characterize40Cr quenched and tempered steel’s dynamic responses andconstitutive model when performing numerical analysis ofhigh-speed cold rolling.

A large number of researches have been carried out inunderstanding and modeling the mechanical behavior of

metals under different strain rates and temperatures in bothexperimental and theoretical schemes. Thermomechanicalresponses of H13 hardened steel over a strain rate rangefrom 103 s−1 to 104 s−1 and a temperature range of 293K to673K was investigated by Lu and He [3]. In their study, thedetermination of the Johnson-Cook material parameters ofH13 hardened steel was based on the experimental results.The Johnson-Cook (JC) constitutive model was shown tocorrelate and predict the observed responses reasonablywell. The quasistatic (10−3 s−1∼10 s−1) and dynamic responses(650 s−1∼8500 s−1) at room temperature of pure iron wereinvestigated by Bao et al. [4]. In their study, the determinationof the Johnson-Cook material parameters of pure iron wasbased on the experimental results. The model predictionswere compared with the experimental results, and a goodagreement had been observed. A systematic study of theresponse of Ti-6Al-4V alloy and advanced high strengthsteels under quasi-static and dynamic loading at differ-ent strain rates and temperatures was investigated by Khanet al. [5, 6]. In their study, the correlations and predic-tions using modified Khan-Huang-Liang (KHL) constitutivemodel were compared with those from Johnson-Cook (JC)

2 ISRNMechanical Engineering

model and experimental observations for this strain rate andtemperature-dependent material. Results showed that KHLmodel correlations and predictions were shown to be muchcloser to the observed responses than the corresponding JCmodel predictions and correlations. Segurado et al. [7] useda physically-based model based on self-consistent homoge-nization behavior to provide a texture-sensitive constitutiveresponse of each material point, within a boundary problemsolved with finite elements (FE). The resulting constitutivebehavior was implemented in the implicit FE code Abaqus.The accuracy of the proposed implicit scheme and the correcttreatment of rotations for prediction of texture evolutionwerebenchmarked by experiment. The potential of the multiscalestrategy was illustrated by a simulation of rolling of a face-centered cubic (FCC) plate. Voyiadjis and Abed [8] devel-oped a coupled temperature and strain rate microstructurephysically based that yield function dynamic deformations ofbody-centered cubic (BCC) metals. Computational aspectsof the model were addressed through the finite elementimplementation with an implicit stress integration algorithm.Numerical implementation for a simple compression prob-lem meshed with one element was used to validate themodel. A physically basedmodel was developed based on themechanism of dislocation kinetics of 3003 Al-Mn alloy overa wide range of strain rates and temperatures by Guo et al.[9]. Comparing the model predictions with the experimentalresults, a good agreement had been observed. The flow stressof 40Cr at various deformations was studied by Li et al. [10].In their study, an improved model suiting for hot and warmworking was developed, but this study was performed over avery limited range of strain rates (0.1 s−1∼10 s−1), which is notsuitable for numerical simulation of high-speed cold rollingforming.Therefore, the corresponding mechanical responsesof 40Cr quenched and tempered steel under different strainrates and temperatures need to be well understood.

In the current research, quasi-static loading (0.004 s−1)at room temperature and dynamic compression experimentsover a strain rates range from 632 s−1 to 5160 s−1 and atemperature range of 293K∼673K of 40Cr quenched andtempered steel were presented. Based on the experimen-tal observations of strain rate and temperature sensitivity,Johnson-Cook (JC) constitutive model was established todescribe the flow stress of 40Cr quenched and tempered steel.Determination of material constants was made. Correlationsand predictions with Johnson-Cook model and experiment-observed responses were analyzed.

2. Experiment

2.1. Experimental Material and Scheme. A 40Cr quenchedand tempered steel was used in the current study. All thespecimens used in this study were made from the same pieceof round bar stock. The specimens were machined usingquenching and tempering, wire EDM, and cutting and grind-ing. The specimen geometry that was used in all the samplestested in this study is Φ 5mm × 4mm. End face of thespecimen’ depth of parallelism is 0.002, and the hardness isHRC 26∼30.

1

23 4

5 6 7 8 9 10 11 12

13 14 15

Figure 1: Schematic of compression split-Hopkinson pressure bar.(1) Outlet valve; (2) inlet valve; (3) back air chamber; (4) frontair chamber; (5) strike bar; (6) incident bar; (7) reaction mass; (8)heating furnace and thermal couple; (9) specimen; (10) transmittedbar; (11) absorption; (12) driver; (13) support; (14) air pipe; and (15)plunger.

In order to analyze the dependence of flow stress on thecoupled effect of strain hardening, strain rate, and temper-ature, five groups of dynamic compression experiments andone group of quasi-static compressionwere performed.Threegroups of dynamic compression experimentswere performedat different strain rates of 1072 s−1, 2534 s−1, and 5160 s−1at room temperature. Two groups of dynamic compressionexperimentswere performed at the temperatures of 473K and673K and at the strain rates of 1193 s−1 and 1380 s−1. Quasi-static compression experiments were performed at the strainrate of 0.004 s−1 at room temperature.

2.2. Experimental Conditions. Quasi-static compression exp-eriment was conducted using the MTS system with anaxial load capacity of 250KN. The mechanical response of40Cr quenched and tempered steel under dynamic loadingwas characterized by using the compression split-hopkinsonpressure Bar (SHPB) technique. The schematic of the exper-imental set up is shown in Figure 1. The system mainlyconsists of a striker bar, incident bar, and transmitted bar.Thediameter and material of the striker bar was the same as thatof incident and transmitted bar. All of them were remainingin a linear elastic state of stress during the test. Incident barand transmitted bar were perfectly aligned.

Split-Hopkinson pressure bar (SHPB) technique satisfiesfour assumptions [11]: (1) fundamental assumption of one-dimensional stress wave propagation theory; (2) having auniform cross section and stress distribution over the entirelength; (3) ignoring friction; and (4) ignoring inertia.

2.3. Experimental Procedures

2.3.1. Quasi-Static Uniaxial Compression Experiments.Quasi-static tensile experiments were performed at roomtemperature. Constant velocity of the testing machine was20mm/min. The specimen-gripping portion is clamped byfriction force between the wall of the fixture and the wall ofthe hardened steel block. A high elongation uniaxial straingage, bonded on the gage section of the specimen, was used tomeasure strain.The strain of this experiment was achieved at31.5%. The load obtained from the MTS load transducer wasused to calculate the stress. The strain rate of this experimentwhich was calculated from the engineering strain and timewas 0.004 s−1. In all the experimental results discussed in

ISRNMechanical Engineering 3

−0.0001 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005−0.075

−0.050

−0.025

0.000

0.025

0.050

0.075

0.100

Volta

ge (V

)

Time (s)

Incident barTransmitted bar

Incident wave

Transmitted wave

Reflected wave

Figure 2: Pulse of 40Cr quenched and tempered steel’ dynamiccompression experiment.

this study, the true stress-strain responses of the material arecalculated from their engineering stress-strain relationship.

2.3.2. Dynamic Compression Experiments at Different StrainRates and Temperatures. The specimen was sandwichedbetween the input and output bars. The required strain rateswere obtained through adjusting the gage lengths of the strikebar and the pressure of the gas gun.Thedynamic compressionexperiment was conducted by pressuring the gas gun andreleasing the solenoid valve. This results in imparting highspeedmotion to the projectile sitting in the gas gun.The highspeed projectile impacts the incident bar generating compres-sion waves in the incident bar and in the projectile. Whenthe stress wave reaches the interface between the incidentbar and the specimen, due to the impedance mismatch at theincident bar-specimen interface, part of the incident pulse isreflected back (reflected pulse) into the incident bar, and therest is transmitted into the specimen. There will be furtherreflections and transmissions within the specimen and acompressive pulse gets transmitted into the transmitter bar(transmitted pulse) (see Figure 2). The strain gages bondedon the incident and transmitted bar measures the elasticdeformation of the bars which were later converted to plasticstrains on the specimen using one-dimensional wave theory.

3. Experimental Results and Discussion

3.1. Strain Rate Sensitivity. The true stress and true straincurves from the uniaxial compression experiments at strainrates of 0.004 s−1, 632 s−1, 1072 s−1, 2534 s−1, and 5160 s−1 andat the temperature of 293K are shown in Figure 3. 40Crquenched and tempered steel showed positive strain-ratesensitivity. Compared with quasi-static experiment, it wasobserved that the flow stress and elastic modulus of 40Crquenched and tempered steel under dynamic condition were

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.400

180

360

540

720

900

1080

1260

1440

1620

1800

True

stre

ss (M

Pa)

True strain

0.004 632 1072

2534 5160

40Cr

s−1 s−1

s−1s−1

s−1

Figure 3: The measured responses of 40Cr under compressionexperiments at different strain rates and at 293K.

significantly increased and that the plastic deformation wasearlier. Under high strain rates, with the increase of strain,the true stress and true strain curves are close to ideal elastic-plastic model. The reason for this phenomenon is that theplastic deformation of 40Cr quenched and tempered steelin high strain rate is of adiabatic nature. Thermal softeningand hardening of the material interact with each other. Thework hardening rate in 40Cr quenched and tempered steelincreased with strain, and at the same time, the work hard-ening rate in 40Cr quenched and tempered steel decreasedwith increase in temperature. In addition, with the increaseof strain, the effect thermal softening played on flow stresswas larger than hardening.

3.2. Temperature Sensitivity. The true stress and true straincurves from the uniaxial compression experiments at tem-peratures of 293K, 473K, and 673K and at strain ratearound 1000 s−1 are shown in Figure 4. 40Cr quenched andtempered steel showed positive temperature sensitivity. Thetrue stress and true strain curves all experience a suddenrise and declining procedure. This illustrates that dramaticgliding happened in the initial deformation of material. Thereason for gliding is the movement of dislocation. Aroundeach 200K increase of temperature, the flow stress of 40Crquenched and tempered steel is of obvious decrease. Thisis because temperature has a pronounced effect on the flowstress. With the temperature increase, capacity for actionof the atom is more active, the movement of dislocation iseasier, and hardening caused by pilling up of dislocations isweaker. So, the resistance of material deformation decreased,while thermal softening increased obviously. The higher thetemperature, themore obvious thermal softening is, themoreflat the curve.


0.00 0.06 0.12 0.18 0.24 0.300

300

600

900

1200

1500

True

stre

ss (M

Pa)

True strain

40Cr

1072 s−1, 293K1193 s−1, 473K1380 s−1, 673K

Figure 4: The measured responses of 40Cr under compressionexperiments at different temperatures and at strain rate around1000 s−1.

4. Constitutive Modeling

A constitutive relation needs to define the dependence offlow stress on the coupled effect of strain hardening, strainrate, and temperature. The flow stress is observed to be afunction of plastic strain, strain rate and temperature. Thesedependencies are usually expressed as multiplicative terms asfollows:

𝜎 (𝜀, 𝜀, 𝑇) = 𝜎

0

(𝜀, 𝜀, 𝑇) 𝑔 ( 𝜀, 𝑇) ℎ (𝑇) . (1)

4.1. Establishment of Constitutive Modeling. Johnson-Cookmodel, which has the capability to mathematically describethe mechanical responses of the material when it is subjectto loadings under different strain rates and temperatures,is a purely phenomenological model. This model is usedfrequently in engineering structures due to its advantage offewer constants, suitability of various crystal structures, andits ability to model the observed material response as closelyas with models with many more constants [12]. Based on theprevious analysis, it can be seen that this model is suitable todescribe the flow stress of 40Cr quenched and tempered steel.

The JC model is given by the following equation:

𝜎 = [𝐴 + 𝐵(𝜀

𝑝

)

𝑛

] (1 + 𝐶 ln 𝜀

∗

) (1 − 𝑇

∗𝑚

) , (2)

where 𝜎 is the stress and 𝜀

𝑝 is the plastic strain. 𝑇, 𝑇𝑚

,and 𝑇

𝑟

are current, melting, and reference temperatures,respectively. The five material constants are 𝐴, 𝐵, 𝑛, 𝐶,and 𝑚. 𝜀

∗

= 𝜀

𝑝

/𝜀

0

is the dimensionless plastic strain rate for𝜀

0

= 0.004 s−1 and 𝑇

∗

= (𝑇−𝑇

𝑟

)/(𝑇

𝑚

−𝑇

𝑟

) is the homologoustemperature for 𝑇

𝑟

= 293K. The expression in the firstset of brackets gives the stress as a function of strain for𝜀

∗

= 1.0 and 𝑇

∗

= 0. The expressions in the second andthird sets of brackets represent the effects of strain rate andtemperature, respectively.

4.2. Determination of Material Constants

4.2.1. Determination of Constant 𝑛 and 𝐵. Strain hardeningexists in the process of 40Cr quenched and tempered steelquasi-static compression experiment. The material constantsof 𝐴, 𝐵, and 𝑛 should be obtained from data of quasi-staticcompression experiment at room temperature. At referencetemperature (𝑇 = 239K) and reference strain rate ( 𝜀 =

0.004), (2) can be expressed as

𝜎 = 𝐴 + 𝐵𝜀

𝑛

. (3)

The constitutive model of 40Cr quenched and temperedsteel at quasi-static and room temperature is (3). Take thelogarithm of both sides of (3):

ln (𝜎 − 𝐴) = ln𝐵 + 𝑛 ln 𝜀. (4)

The relationship between ln(𝜎 − 𝐴) and ln 𝜀 can be givenby (4). ln𝐵 is the intercept and 𝑛 is the slope of the fitting linein the ln(𝜎 − 𝐴) − ln 𝜀 plot. The value of 𝐴 is calculated fromthe yield stress 𝜎

0.085

= 905MPa.The value of 𝑛 and 𝐵 can beobtained as 0.26 and 226MPa from the fitting of the workhardening experiment data under quasi-static condition atroom temperature.

4.2.2. Determination of Constant 𝐶. In order to obtain thematerial constant of 𝐶, data of dynamic compression experi-ments are needed. At reference temperature, there is no flowsoftening term as 𝑇∗ = 0, and (2) can be expressed as

𝜎 = (𝐴 + 𝐵𝜀

𝑛

) (1 + 𝐶 ln 𝜀

∗

) . (5)

Further expressed as

𝜎

𝐴 + 𝐵𝜀

𝑛

= 1 + 𝐶 ln 𝜀

∗

. (6)

The relationship between 𝜎/(𝐴 + 𝐵𝜀

𝑛

) and ln 𝜀

∗ can beobtained.Thematerial constant𝐶 can be evaluated as 0.03 bylinear fitting of the experiment data. These data are the truestress at the strain rates of 632 s−1, 1072 s−1, and 5167 s−1. Foreach true stress, the true strain is 0.1.

4.2.3. Determination of Constant 𝑚. Thermal softening con-stant means the dependence of flow stress on temperature.The determination of constant 𝑚 needs the dynamic experi-ment data at elevated temperatures. At reference temperatureand reference strain rate, take the logarithm of both sides of(2), it can be expressed as:

ln[1 −

𝜎

(𝐴 + 𝐵𝜀

𝑛

) (1 + 𝐶 ln ( 𝜀/𝜀

0

))

] = 𝑚 ln𝑇

∗

. (7)

It can be seen that𝑚 is the slope of the straight, and𝑚 canbe obtained as 0.83 by linear fitting of the experiment data.These data are the true stress at the strain rates of 1124 s−1,1193 s−1, 1236 s−1 and 1380 s−1 at the temperatures of 373K,473K, 573K, and 673K. For each true stress, the true strain is0.1.

ISRNMechanical Engineering 5

By using the obtained constants of Johnson-Cook consti-tutivemodel, flow stress of 40Cr quenched and tempered steelcan be expressed as

𝜎 = (905 + 226𝜀

0.21

) (1 + 0.03 ln 𝜀

0.004

) [1 − (𝑇

∗

)

0.83

] .

(8)

4.3. Correlations with the Constitutive Model. As for the dyn-amic data of 40Cr quenched and tempered steel, thermalsoftening from the adiabatic deformation was effectivelyconsidered by converting the increment of temperature fromthe stress-strain curve using following equation:

Δ𝑇 =

𝛽

𝜌𝐶

𝑝

∫

𝜀

𝑝

0

𝜎 (𝜀

𝑝

) 𝑑𝜀

𝑝

, (9)

where 𝛽, 𝜌(7.87 kg/m3), 𝐶𝑝

[470𝐽/(kg ⋅ k)] are the fractionof heat dissipation caused by the plastic deformation, massdensity, and specific heat at constant pressure, respectively.The value of 𝑏 is normally assumed as 0.9 for metals.

Use of the Johnson-Cook constitutive model defined in(2) allows (9) to be written as

Δ𝑇 =

𝛽

𝜌𝐶

𝑝

∫

𝜀

𝑝

0

𝜎 (𝜀

𝑝

) 𝑑𝜀

𝑝

=

𝛽

𝜌𝐶

𝑝

∫

𝜀

𝑝

0

[𝐴 + 𝐵(𝜀

𝑝

)

𝑛

] [1 + 𝐶 ln ( 𝜀)

∗

] [1 − (𝑇

∗

)

𝑚

] 𝑑𝜀

𝑝

=

𝛽𝜀

𝑝

𝜌𝐶

𝑝

[𝐴 +

𝐵

𝑛 + 1

(𝜀

𝑝

)

𝑛+1

] [1 + 𝐶 ln ( 𝜀)

∗

] [1 − (𝑇

∗

)

𝑚

] .

(10)

Substituting temperature increment (10) with parametersresults in the temperature increment (10) taking the followingform:

Δ𝑇 =

0.9𝜀

𝑝

0.787 × 4.7

[905 +

226

0.26 + 1

(𝜀

𝑝

)

0.26+1

]

× [1 + 0.03 ln ( 𝜀)

∗

] [1 − (𝑇

∗

)

0.83

] .

(11)

The correlations of flow stress and temperature increaseduring a compression test at a nominal strain rate of 2534 s−1at room temperature are shown in Figure 5. It can be observedthat the correlations and prediction from the model wereclose to the experimental results. This provided a theoreticalbasis and material parameters for further numerical simula-tion of coupled thermo-mechanical finite element models.

The JC model correlations of the experimental dataobtained at different temperatures and at strain rate around1000 s−1 is shown in Figure 6. It can be observed that theflow stress level as well as the work hardening rate decreasescontinuously with the increase of temperature. Further, thepredictions from the constitutive model were in good agree-ment with the experimental results.

0.04 0.08 0.12 0.16 0.20 0.24 0.280

300

600

900

1200

1500

1800

JC modelExperiment

True strain

True stress

Tem

pera

ture

incr

ease

(K)

0

30

60

90

120

150

JC modelExperiment

Temperature increase

Figure 5: Flow stress and temperature increase during a compres-sion test at a nominal strain rate of 2534 s−1 at room temperature.

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.2500

200

400

600

800

1000

1200

1400

1600

1800Tr

ue st

ress

(MPa

)

True strain

1072 s−1 293K experiment1072 s−1 293K JC model1193 s−1 473K experiment1193 s−1 473K JC model

1380 s−1 673K experiment1380 s−1 673K JC model

Figure 6: The JC model correlations of the experimental dataobtained at different temperatures and at strain rate around 1000 s−1.

5. Conclusions

(1) Comparing dynamic experiment with quasi-staticexperiment, it can be observed that the flow stressunder dynamic condition is much higher. 40Crquenched and tempered steel showed positive strain-rate sensitivity.

(2) It can be observed fromdynamic experiments that theflow stress 40Cr quenched and tempered steel showspositive strain-rate and temperature sensitivity. Theflow stress increase with the strain rates increase, and


decrease with the increase of temperature simultane-ously. Further, the flow stress showed positive thermalsoftening at elevated temperatures.

(3) The time of the dynamic compression experimentsfor 40Cr quenched and tempered steel was short,and 90% of heat dissipation was caused by theplastic deformation, which resulted in a temperatureincrease in specimen. The highest temperature at thestrain rate of 2534 s−1 was around 100K. Thermalsoftening of material increased with the increase ofstrain and strain rates. The true stress and true straincurve at the strain rate of 5160 s−1 was close to idealelastic-plastic model.

(4) It can be observed that the correlations and predictionfrom themodel were close to the experimental results.Johnson-Cook can preferably describe the effect ofstrain hardening, strain rate and temperature of40Cr quenched and tempered steel. This provided atheoretical basis and material parameters for furthernumerical simulation of coupled thermo-mechanicalfinite element models.

Acknowledgments

The authors acknowledge the financial support of NationalNatural Science Foundation of China (51075124), (50975229),and the Aviation Science Fund (2009ZE54010).

References

[1] J. Quan, F. Cui, J. Yang, H. Xu, and Y. Xue, “Numerical simu-lation of involute spline shaft’s cold-rolling forming based onANSYS/LS-DYNA,” China Mechanical Engineering, vol. 19, no.4, pp. 419–427, 2008.

[2] L. Yan, Y. Mingshun, Y. Qilong, and Z. Jianming, “Study onthe metal flowing of lead screw cold roll-beating forming,”Advanced Science Letters, vol. 4, no. 6-7, pp. 1918–1922, 2011.

[3] S. Lu and N. He, “Experimental investigation of the dynamicbehavior of hardened steel in high strain rate and parameterfitting of constitutive equation,” China Mechanical Engineering,vol. 19, no. 19, pp. 2382–2385, 2008.

[4] W.-P. Bao, X.-P. Ren, and Y. Zhang, “The characteristics of flowstress and dynamic constitutive model at high strain rates forpure iron,” Journal of Plasticity Engineering, vol. 16, no. 5, pp.125–129, 2009.

[5] A. S. Khan,M. Baig, S.-H. Choi, H.-S. Yang, and X. Sun, “Quasi-static and dynamic responses of advanced high strength steels:experiments and modeling,” International Journal of Plasticity,vol. 30-31, pp. 1–17, 2012.

[6] A. S. Khan, Y. S. Suh, and R. Kazmi, “Quasi-static and dynamicloading responses and constitutivemodeling of titaniumalloys,”International Journal of Plasticity, vol. 20, no. 12, pp. 2233–2248,2004.

[7] J. Segurado, R. A. Lebensohn, J. Llorca, and C. N. Tome, “Mul-tiscale modeling of plasticity based on embedding the vis-coplastic self-consistent formulation in implicit finite elements,”International Journal of Plasticity, vol. 28, no. 1, pp. 124–140,2012.

[8] G. Z. Voyiadjis and F. H. Abed, “A coupled temperature andstrain rate dependent yield function for dynamic deformationsof bcc metals,” International Journal of Plasticity, vol. 22, no. 8,pp. 1398–1431, 2006.

[9] W. G. Guo, X. Q. Zhang, J. Su, Y. Su, Z. Y. Zeng, and X. J. Shao,“The characteristics of plastic flowand a physically-basedmodelfor 3003 Al-Mn alloy upon a wide range of strain rates andtemperatures,” European Journal of Mechanics, A/Solids, vol. 30,no. 1, pp. 54–62, 2011.

[10] X. Li, H.-B. Zhang, X.-Y. Ruan, Z.-H. Luo, and Y. Zhang, “Anal-ysis and modeling of flow stress of 40Cr steel,” Cailiao Gong-cheng/Journal of Materials Engineering, no. 11, pp. 41–49, 2004.

[11] M. A. Kariem, J. H. Beynon, and D. Ruan, “Misalignment effectin the split Hopkinson pressure bar technique,” InternationalJournal of Impact Engineering, vol. 47, pp. 60–70, 2012.

[12] G. R. Johnson and W. H. Cook, “A constitutive model and datafor metals subjected to large strains, high strain rates and hightemperatures,” in Proceedings of the 7th International Sympo-sium on Ballistic, p. 541, The Hague, The Netherlands, 1983.

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